Question: A circle has a sector with area $21\pi$ and central angle $\dfrac{7}{6}\pi$ radian. What is the area of the circle? ${36\pi}$ $\color{#9D38BD}{\dfrac{7}{6}\pi}$ ${21\pi}$
The ratio between the sector's central angle $\theta$ and $2 \pi$ radians is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{2 \pi} = \dfrac{A_s}{A_c}$ $\dfrac{7}{6}\pi \div 2 \pi = 21\pi \div A_c$ $\dfrac{7}{12} = 21\pi \div A_c$ $A_c \times \dfrac{7}{12} = 21\pi$ $A_c = 21\pi \times \dfrac{12}{7}$ $A_c = 36\pi$